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Global optimization of mass and property integration networks with in-plant property interceptors Fabricio Na´poles-Rivera a, Jose´ Marı´a Ponce-Ortega b, Mahmoud M. El-Halwagi c, Arturo Jime´nez-Gutie´rrez a, a
´logico de Celaya, Celaya Gto, Mexico Chemical Engineering Department, Instituto Tecno ´s de Hidalgo, Morelia, Mich, Mexico Chemical Engineering Department, Universidad Michoacana de San Nicola c Chemical Engineering Department, Texas A&M University, College Station, TX, USA b
a r t i c l e in fo
abstract
Article history: Received 10 October 2009 Received in revised form 20 February 2010 Accepted 29 March 2010 Available online 1 April 2010
This paper presents a mathematical programming model for the optimal design of mass and property integration networks that include property interceptors within the structure of the network, as opposed to the end-of-pipe use of such interceptors. The model is based on a recycle and reuse scheme that simultaneously satisfies process and environmental constraints. The properties considered in this work are composition, toxicity, theoretical oxygen demand, pH, density and viscosity. The property mixing rules included in the model give rise to bilinear terms for the property operators, and a global optimization algorithm is used for the solution of the model. The model minimizes the total annual cost of the network, which includes the fresh sources cost and the annualized property treatment system and the piping costs. Three examples are included to show the applicability and advantages of the proposed model. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Mass integration Property integration Mixed integer nonlinear programming Mathematical modeling Optimization Chemical processes Design
1. Introduction During the last three decades, the exponential growth in population and industrialization has affected the massive use of natural resources to the point where their availability and their cost have become major challenges for sustainability. It is thus important to develop strategies to optimize the industrial use of natural resources. Also, environmental constraints have become tighter, which should affect the design of new processes or the modification of existing ones. In this context, mass integration strategies have been effectively utilized to face both economic and environmental problems. Usually, mass integration strategies aim to optimize the allocation, transformation, and separation of species and streams. An important class of mass integration problems deals with the synthesis of networks that allow the recycle and reuse of process sources to minimize the consumption of fresh sources and waste discharges. The mass integration techniques for recycle/reuse can be classified into two categories (e.g., Pillai and Bandyopahyay, 2007; El-Halwagi, 2006), methodologies that use the principles of pinch analysis and methods based on mathematical programming
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[email protected] (A. Jime´nez-Gutie´rrez). 0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.03.051
techniques. For a recent survey on pinch analysis techniques see the review paper by Foo (2009). Works based on mathematical programming techniques include the one by Takama et al. (1980), who addressed the water allocation problem in a petroleum refinery. This model considered all possible configurations between wastewatertreating units and water-using units in order to minimize both fresh water consumption and waste generation. The problem was treated with a two-level formulation in which only the upper level or allocation problem was solved. Quesada and Grossmann (1995) showed a model to obtain mass exchange networks when mass balances give rise to bilinear terms. To deal with the bilinear terms, they used a linear reformulation in order to obtain a valid lower bound to the optimal solution, and then applied a spatial branch and bound search to obtain the global optimum structure. Galan and Grossmann (1998) extended the model to include the selection of water treatment technologies. Lee and Grossmann (2003) developed a generalized disjunctive programming problem based on the model by Quesada and Grossmann (1995), taking as discrete choices the existence or not existence of process units. Karuppiah and Grossmann (2006) proposed a modification to the global optimization method by Quesada and Grossmann (1995) in order to improve the lower bounding step. They applied this new procedure for the synthesis of integrated water networks. Alva-Argaez et al. (1998) tackled the problem of waste
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water minimization by developing a superstructure that contained all the possibilities for water recycle, reuse and regeneration. The resulting model was a mixed integer nonlinear programming problem (MINLP) with bilinear terms. To avoid the complexities associated to the solution of the nonconvex MINLP, they decomposed it into a sequence of MILP problems to provide a near optimal solution. Huang et al. (1999) developed a superstructure for the water allocation problem, which was based on the one presented by Takama et al. (1980) but it contained new features such as the possibility to handle multiple sources and sinks. Additionally, it included the design of the treatment system so that the optimization process could not only minimize the amount of used water, but also the waste treatment capacity. Gabriel and El-Halwagi (2005) addressed the problem of the synthesis of mass exchange networks using a source-interception-sink representation with a simplified linear programming
model. Ng et al. (2007a, 2007b) addressed the problem of maximum water recovery using a recycle/reuse scheme. Once the maximum water recovery policy has been accomplished, a regeneration system is used to meet environmental regulations over the load of contaminant. Lancu et al. (2009) tackled the problem of wastewater minimization for the multi-contaminant case through the regeneration of streams. They proposed the use of in-plant regeneration systems to reduce the amount of water sent to disposal. Hul et al. (2007), Tan et al. (2008), and Chakroborty (2009) have recently reported other works in this field. It should be noted that many of the reported methods on the synthesis of recycle/reuse networks have ignored the fact that environmental and process constraints do not depend only on compositions and flows of the streams, but also on properties such as pH, density, viscosity, toxicity, color, and theoretical
max ψ pmin , j ≤ ψ p, j ≤ ψ p, j
z min ≤ z j ≤ z max j j r 1
1
2
2
Fresh
r
Sinks NSinks
NSinks
z environment ≤ z max j j n x ψ pm,ienvironment ≤ ψ p ,environment ≤ ψ pm,aenvironment
Fig. 1. Source–sink representation for mass and property integration including waste treatment.
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oxygen demand. Shelley and El-Halwagi (2000) introduced the notion of clusters and used it as a basis for a component-less design method based on tracking functionalities and properties, rather than tracking their individual components. They also showed the possibility to apply this new concept in a way analogous to mass integration problems, as part of a framework that El-Halwagi et al. (2004) called property integration defined as a functionality-based holistic approach to the allocation and manipulation of streams and processing units, which is based on the tracking, adjustment, assignment, and matching of functionalities throughout the process. Property integration methodologies can be classified into graphical, algebraic, and optimization techniques. Graphical techniques are suitable when up to three properties are considered. Works showing such techniques have been developed by Shelley and El-Halwagi (2000), Eden et al. (2002), El-Halwagi et al. (2004), El-Halwagi and Kazantzi (2005), Kazantzi et al. (2007), and Eljack et al. (2008). When more than three properties are considered, algebraic and optimization-based tools are required (see for example the works by Qin et al., 2004; Grooms et al., 2005; Foo et al., 2006; Ng et al., 2008, 2009; Ponce-Ortega et al., 2009, 2010). As a consequence of not including environmental constraints into the model formulation, the network that minimizes fresh water consumption or waste discharge minimization may be considered as an incomplete solution to the overall problem. It is possible that the additional waste treatment system needed to meet the environmental constraints can affect significantly the total cost of the complete network and yield a suboptimal solution. Ponce-Ortega et al. (2009) found that for a direct recycle scheme such as the one shown in Fig. 1, the simultaneous consideration of process and environmental constraints reduces the total annual cost of the network compared to a solution given by a sequential approach (i.e. the optimization of the mass exchange network followed by the final treatment of waste streams). Although the strategy proposed by Ponce-Ortega et al. (2009) considers a simultaneous approach, it assumes a centralized waste treatment plant, which implies that all the waste sources are collected and treated in a common facility. Galan and Grossmann (1998) found that the use of a distributed waste treatment system or recycle reuse network scheme (see Fig. 2) can reduce the flow rate to be processed in each treatment
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unit, therefore reducing the treatment unit cost. Ponce-Ortega et al. (2010) have recently proposed a simplified formulation for the recycle and reuse scheme; a conceptual model was used to avoid the mixing of streams, thus preventing the existence of bilinear products in the property balances around the mixing points. Their superstructure formulation can be seen in Fig. 3. In this paper, a rigorous mathematical programming model is presented for the mass and property integration problem under a recycle and reuse configuration. Fig. 4 shows the model structure. The model includes an in-plant property treatment system as opposed to an end-of-pipe treatment facility, so that the waste is treated as part of the mass integration network. Both process and environmental constraints for properties such as toxicity, THOD, pH, viscosity and density are considered; mass and composition constraints are also included. The process sources can be segregated and sent to any treatment unit. It is assumed that only one property can be treated within each unit, and the outlet flow of each treatment unit can be sent either to the process units, waste or to treat a different property; the formulation prevents that a stream returns to the same unit or to a unit in which it has already been treated. The possibility to mix streams with unknown flows and properties gives rise to bilinear terms in the property balances as part of the MINLP problem.
2. Outline of the proposed model The problem addressed in this paper consists of finding the optimal mass and property integration network under a recycle– reuse scheme that minimizes the total annual cost of the overall process while satisfying simultaneously process and environmental constraints. Such constraints are known prior to the optimization process. Also known are the process and fresh sources available, with given flow rates, compositions and properties. Additionally, the costs of piping and fresh sources, and the cost and efficiency of each available interceptor to treat the stream properties are known. The sets used in the model formulation are defined as follows. NSINKS contains the sinks for the streams within the process, while NSOURCES and FRESH contain the process and fresh sources, and NPROP contains the properties constrained by process sinks or
Sinks
Sources
Property constraints max Pp,min j ≤ Pp, j ≤ Pp, j i=1
j=1 G1, Pp, 1 j=2 G 2, P p, 2
...
i=2
...
Property Interception Network
j = NSinks GNSinks, Pp, NSinks
i = Nsources Waste
Environmental constraints max
Pp,min environment ≤ Pp, environment ≤ Pp, environment
Fig. 2. Recycle and reuse network scheme.
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...
Fresh sources r=1
r=N fresh Sinks Property constraints min
max
...
Pp, j ≤ Pp, j ≤ Pp, j j=1 G1, Pp, 1
Process sources
...
...
...
...
...
i=1
...
...
j = NSinks GNSinks, Pp, NSinks
Waste
...
i = Nsources ...
Environment constraints max
...
Pp,min environment ≤ Pp, environment ≤ Pp, environment
Fig. 3. Simplified formulation for the recycle and reuse mass and property integration.
by environmental regulations. NUNINTS accounts for all of the treatment units and NINT contains the available units to treat each property. The variables Fr, fr,j, Wi, wi,u, d1u,u , mu, Gj and gu,j refer to the flow rates across the network (see Fig. 4). Property operators cp(P) are used; these operators are based on proper mixing rules for each property, as reported in Table 1. The properties considered in this paper are composition, toxicity, THOD, pH, density and viscosity. Properties are identified as InSource InUnit is the value for property p in source i, Pp,u and follows. Pp,i OutUnit are the values at the inlet and at the outlet of treatment Pp,u InFresh InSink is the value in fresh source r, and Pp,j is the unit u, Pp,r value for entering sink j. Hy is the operation time per year (taken
here as 8000 h/yr), CostrFresh is the cost per pound of fresh source r that is used. CostuUnit is the cost per pound treated in unit u. For Int piping costs, pipIn i,u is the cost from source i to unit u, pipu,u1 from InFresh from unit u to sink j, and pip from unit u to unit u1, pipExit u,j r,j fresh source r to sink j (jawaste). TAC is the total annual cost of the network.
3. Model formulation The model is based on the configuration shown in Fig. 4, and consists of mass balances at mixing and splitting points of the
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Fig. 4. Rigorous mass and property integration network: recycle and reuse scheme.
Splitting of process sources: Process sources are segregated and sent to any of the treatment units to treat one property at a time: X wi,u , i A NSOURCES ð2Þ Wi ¼
Table 1 Some mixing property operators. Property
Operator
Composition Toxicity Chemical oxygen demand pH
cz ðzÞ ¼ z cTox ðToxÞ ¼ Tox cCOD ðCODÞ ¼ COD cpH ðpHÞ ¼ 10pH
Density
cr ðrÞ ¼
u A NUNINTS
1
r
Viscosity
cm ðmÞ ¼ logðmÞ
Reid vapor pressure Electric resistivity
cRVP ðRVPÞ ¼ RVP 1:44
Paper reflectivity
cR1 ðR1 Þ ¼ R5:92 1 cColor ðColorÞ ¼ Color cOdor ðOdorÞ ¼ Odor
Color Odor
cR ðRÞ ¼
1 R
process, property balances in each mixing point, process and environmental constraints, and logic disjunctions to choose which interceptor should be used for each property adjustment that is needed. The disjunctions are reformulated using the convex hull formulation. The objective function is set as the minimization of the total annual cost of the network. Splitting of fresh sources: Fresh sources can be sent to any sink to complement, if needed, the treatment capabilities of process sources to meet process constraints: Fr ¼
X
fr,j ,
r A FRESH
j A NSINKS j a waste
Notice that the fresh sources cannot be sent to the waste.
ð1Þ
Mass balance at the inlet of the property interceptors: A recycle and reuse strategy is used; hence, a mass balance in the mixing point prior to any interceptor indicates that the inlet flow should be equal to the flow sent from the process sources plus the flow coming from any other interceptor: X X wi,u þ du1 ,u ¼ mu , u A NUNITS ð3Þ u1 A NUNITS u1 a u
i A NSOURCE
This scheme allows the treatment of more than one property for the process sources. The model has an additional restriction that avoids the possibility of a redundant recirculation, which means that a source cannot return to a treatment unit in which it has already been used. Property tracking at the inlet of the property interceptors: To determine the properties at the inlet of any interceptor, the following property mixing rule is used: i i X h X h InSource OutUnit þ wi,u cp Pp,i du1 ,u cp Pp,u 1 i A NSOURCE
¼ mu cp
In Unit , Pp,u
u1 A NUNITS u1 a u
u A NUNITS, p A NPROP
ð4Þ
One must take into account that each property has a different mixing operator cp. Table 1 presents a list of mixing property operators for some common properties. As can be seen in Eq. (4), the recycle and reuse configuration gives rise to bilinear terms, since both the mass recirculated and its properties are unknown. Also unknown are the total flow entering the interceptors and its property values.
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Mass balances in the splitting point at the exit of the interceptors: The flow rate at the exit of any treatment unit is equal to the flow rate sent to the process sinks plus the flow rate sent to any other interceptor: X X gu,j þ du,u1 , u A NUNITS ð5Þ mu ¼ j A SINKS
u1 A NUNITS u1 a u
Mass balances at the mixing point before the sinks: The flow entering a process sink is equal to the sum of the flows coming from the interceptors plus the flow rate from fresh sources: X X fr,j þ gu,j , j A NSINKS ð6Þ Gj ¼ r A FRESH
u A NUNITS
Property tracking at the mixing point before the sinks: A property mixing rule is needed in tracking properties to determine the properties at the entrance of the process sinks. Notice that this balance also contains bilinear terms, given by the product of the flow coming from the treatment units and its property operator, and the product of the waste flow and its property operator: InSink ¼ Gj cp Pp,j
X h
i
InFresh fr,j þ cp Pp,r
r A FRESH
X
h
properties not treated is simply given by
cp0 PpOutUnit ¼ cp0 PpInUnit Uðp0 Þ A NUNITS, 8p a p0 0 ,Uðp0 Þ 0 ,Uðp0 Þ ,
ð10Þ
The disjunction is applied to all properties treated in the property interception network, and the number of interceptors to treat each property is known before the optimization process. All Uðp0 Þ , the terms in the disjunction are linear since the efficiency, aIðUÞ Uðp0 Þ and unit cost, CostIðUÞ , for the interceptors are given. The convex hull formulation (Raman and Grossmann, 1994) is used to model the disjunction as follows. To select the unit to treat the property, the following equation is used: X Uðp0 Þ yIðUÞ ¼ 1, Uðp0 Þ A NUNITS ð11Þ IðUÞ
Then, the optimization variables involved in the disjunction are disaggregated as follows: X cp0 PpOutUnit cp0 pOutUnit,IðUÞ ð12Þ ¼ , Uðp0 Þ A NUNITS 0 ,Uðp0 Þ p0 ,Uðp0 Þ IðUÞ
i OutUnit , gu,j cp Pp,u
cp0 PpInUnit ¼ 0 ,Uðp0 Þ
u A NUNITS
X
cp0 pInUnit,IðUÞ , Uðp0 Þ A NUNITS p0 ,Uðp0 Þ
ð13Þ
IðUÞ
j A NSINKS, p A NPROP
ð7Þ mUðp0 Þ ¼
Constraints: A set of constraints for the process sinks and one for the waste discharged to the environment are needed; these constraints include maximum and minimum properties allowed because of process limits or environmental regulations. For the process sinks: min Sink inSink max Sink Pp,j r Pp,j rPp,j ,
j A NSINK, j awaste, P A NPROP
ð8Þ
For the waste discharged to the environment: min Env max Env Pp,j r Pp,waste rPp,j ,
j ¼ waste, P A NPROP
X IðUÞ mUðp0 Þ ,
Uðp0 Þ A NUNITS
ð14Þ
IðUÞ unit CostUðp 0Þ ¼
X unit,IðUÞ CostUðp , 0Þ
Uðp0 Þ A NUNITS
ð15Þ
IðUÞ
The equations in the disjunction are rewritten in terms of the disaggregated variables: 0Þ ¼ cp0 ppInUnit,IðUÞ 1apUðp cp0 ppOutUnit,IðUÞ 0 ,Uðp0 Þ 0 ,Uðp0 Þ 0 ,IðUÞ , Uðp0 Þ A NUNITS, IðUÞ A NINT
ð16Þ
ð9Þ
Logic disjunctions: To treat each property, a set of interceptors with given efficiencies and unit costs is available. Each interceptor is associated with a certain extent of changing the value of the property. Therefore, as part of the optimization strategy, selection has to be made between a high efficiency—high cost unit, or a lower efficiency—lower cost unit; the best choice can vary for each problem. To model this decision, we use the following disjunction: 3 2 UðpÞ YIðUÞ 6 7 7 6 OutUnit InUnit 7 6 cp Pp,UðpÞ ¼ cP Pp,UðpÞ 1aUðpÞ 8 IðUÞ 7 6 IðUÞ A NINT 4 5 UðpÞ unit ¼ CostIðUÞ mUðpÞ Costp,UðpÞ
0
unit,IðUÞ Uðp Þ IðUÞ ¼ CostIðUÞ mUðp0 Þ , CostUðp 0Þ
Uðp0 Þ A NUNITS, IðUÞ A NINT
ð17Þ
Finally, upper limits for the disaggregated variables are needed: Uðp0 Þ r M cp0 yIðUÞ cp0 ppOutUnit,IðUÞ , Uðp0 Þ A NUNITS, IðUÞ A NINT ð18Þ 0 ,Uðp0 Þ
0
Uðp Þ cp0 pInUnit,IðUÞ , Uðp0 Þ A NUNITS, IðUÞ A NINT rM cp0 yIðUÞ p0 ,Uðp0 Þ 0
IðUÞ m Uðp Þ mUðp 0 Þ rM yIðUÞ ,
Uðp0 Þ A NUNITS, IðUÞ A NINT 0
Unit,IðUÞ Uðp Þ CostUðp r M Cost yIðUÞ , 0Þ
ð19Þ ð20Þ
Uðp0 Þ A NUNITS, IðUÞ A NINT
ð21Þ
0
Uðp Þ where YIðUÞ are logical variables, used to denote that the outlet
property and the treatment cost depend on the unit selected; the property treated and the unit used for treatment are denoted as p0 and uðp0 Þ, respectively, and the set of interceptors available to treat this property is given by I(U). This disjunction implies that when the
Table 2 Efficiency and unit cost per pound of total flow treated for each treatment unit. Property
Interceptor
Efficiency, a
Unit cost ($/lb)
0.98 0.85 1.00 0.90 0.80 0.55 0.99 0.9 99 9 0
0.0065 0.0033 0.0098 0.0075 0.0065 0.0032 0.0063 0.0032 0.0065 0.0034 0
0
Uðp Þ is true (i.e., the associated binary variable Boolean variable YIðUÞ 0
0
0
Uðp Þ Uðp Þ Uðp Þ yIðUÞ is equal to 1) the unit with cost CostIðUÞ and efficiency aIðUÞ
will be selected; notice that only one unit can be selected for each property, depending on the process requirements. If the unit is not
Component Toxicity THOD
0
Uðp Þ is false (i.e., the associated selected, the Boolean variable YIðUÞ 0
Uðp Þ binary variable yIðUÞ is equal to 0), in which case the property does
not change and therefore no additional cost is included. It should be stressed that only one property can be treated by each interceptor. Then, the property balance in each unit for the
pH
None
1
REC REC2 TOX1 TOX2 AER1 AER2 PH1 PH2 POH1 POH2 NONE
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These limits can be set from physical restrictions or from available process data. It is important to stress that an additional unit with efficiency and cost equal to zero has been included. This fictitious unit is
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used for modeling purposes (represented as unit NONE in Fig. 4), and its only task is the mixing of streams. The assumption that only one property can be treated within each unit implies that the rest of the properties in that unit
Table 3 Process (W) and fresh (F) sources characteristics for each example. Stream
Flow rate (lb/h)
Composition (ppm)
Toxicity (%)
Example 1 W1 W2 F1
2900 2450 –
0.033 0.022 0
0.8 0.5 0
Example 2 W1 W2 W3 F1 F2
8083 3900 3279 – –
0.016 0.024 0.22 0 0.01
0.3 0.5 1.5 0 0.1
Example 3 W1 W2 W3 W4 W5 W6 F1 F2 F3
3100 1800 1750 2000 1300 1400 – – –
0.16 0.1 0.11 0.12 0.09 0.2 0 0.01 0.09
0.4 0.7 1.3 0.8 0.4 1.5 0 0.1 0.5
THOD (mg O2/l)
pH
Density (lb/l)
Viscosity (cP)
75 88 0
5.3 5.1 7.0
2.000 2.208 2.204
1.256 1.220 1.002
0.187 48.85 92.10 0 0.01
6.4 5.1 4.8 7.0 6.8
2.000 2.208 2.305 2.204 2.209
1.256 1.220 1.261 1.002 0.992
4.4 3.9 4.7 4.7 3.8 5.7 7.0 6.8 7.1
2.100 2.208 2.305 2.105 2.305 2.102 2.204 2.209 2.215
1.236 1.220 1.241 1.256 1.260 1.259 1.002 0.992 0.988
95 85 90 100 100 100 0 0.00 0.00
Table 4 Process and environmental constraints for each example. Sink Example 1 Maximum 1 2 Waste Minimum 1 2 Waste Example 2 Maximum 1 2 3 Waste Minimum 1 2 3 Waste Example 3 Maximum 1 2 3 4 5 6 Waste Minimum 1 2 3 4 5 6 Waste
Flow rate (lb/h)
Composition (ppm)
Toxicity (%)
3000 1900 –
0.013 0.011 0.005
2 2 0
3000 1900 –
0 0 0
6000 2490 3287 –
THOD (mg O2/l)
pH
Density (lb/l)
Viscosity (cP)
75 75 75
8.0 7.9 9.0
2.8 2.5 3.0
1.202 1.430 2.000
0 0 0
0 0 0
5.9 5.7 5.8
1.8 1.7 1.0
0.871 0.782 1.000
0.013 0.011 0.1 0.005
2 2 2 0
75 100 100 75
8.0 7.8 8.2 9.0
2.8 2.5 2.9 3
1.202 1.430 1.260 2
6000 2490 3287 –
0 0 0 0
0 0 0 0
0 0 0 0
5.3 5.4 5.2 5.5
1.8 1.7 1.85 1.0
0.871 0.782 0.775 1.000
1900 1600 2800 2000 1800 1000 –
0.1 0.01 0.04 0.02 0.01 0.01 0.005
2 2 2 2 2 2 0
100 100 100 100 100 100 50
8.0 7.9 8.1 8.0 8.0 8.0 8.0
2.8 2.5 2.9 2.8 2.7 2.7 2.7
1.292 1.280 1.270 1.291 1.281 1.281 1.290
1900 1600 2800 2000 1800 1000 –
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
5.8 5.5 5.4 5.5 5.5 5.5 5.4
1.8 1.7 1.85 1.75 1.85 1.85 1.00
0.871 0.782 0.775 0.84 0.85 0.85 1.00
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remain unchanged, as shown in Eq. (10). Nevertheless, the model could be easily adapted to consider treatment units capable of treating more than one property at a time; in that case, the efficiency parameter for each property would be known prior to the optimization process. If more than one property could be treated in the same unit, the disjunction here developed and its reformulation would still be valid, such that p would be the set of properties that change and p0 the set of properties that would not change within the unit. This type of information would have to come from experimental data, and would be part of the design data prior to the model development. Objective function: The objective function consists of the minimization of the total annual cost, which includes the cost for the fresh sources, the cost for the interceptors to treat the process sources, and piping costs: X X CostrFresh Fr þ HY CostuUnit Min TAC ¼ HY r A FRESH
þHY
X
u A NUNITS
X
In pipi,u wi,u
u A NUNITSi A NSOURCES
þHY
X
X
pipInt u,u1 du,u1
u A NUNITSu1 A NUNITS
þHY
X
X
pipExit u,j gu,j
u A NUNITSj A NSINKS
þHY
X
X
pipInFresh fr,j r,j
ð22Þ
r A NFRESHj A NSINKS
Reformulation and linearization (Relaxed Model): The existence of bilinear terms makes this model a nonconvex MINLP problem; to find the best possible solution, we use a deterministic global optimization technique similar to the one proposed by Quesada and Grossmann (1995) and improved by Karuppiah and Grossmann (2006). First, a linear programming relaxation of the nonlinear model is constructed by replacing every nonconvex term (bilinear term) by a new variable; in this case i
xip ¼ F i cp
Table 5 Flow rates obtained in the root node of the optimization algorithm. Variable
Rigorous model flow (lb/h)
Relaxed model flow (lb/h)
Difference
F1 f1.1 w1.COMP w1.TOX w1.THOD w1.PH w1.NONE w2.THOD w2.PH W2.NONE dCOMP.TOX dTOX.POH mCOMP mTOX mTHOD mPH mPOH mNONE gCOMP.1 gCOMP.2 gTOX.3 gTHOD.1 gTHOD.2 gTHOD.3 gPH.1 gPH.2 gPOH.2 gPOH.3 gNONE.1 gNONE.2 gNONE.3 GS.1 GS.2 Gwaste
358.7 358.7 2648.6 112.7 138.6 0 0 70.997 0 2379 727.64 49.306 2648.60 840.4 209.6 0 49.306 2379 895.33 1025.6 791.12 0 209.60 0 0 0 31.641 17.665 1745.86 633.13 0 3000 1900 808.79
0 0 2000 0 0 358.8 541.17 871.3 119.86 1458.8 0 0 2000 0 871.307 478.6 0 2000 1000 1000 0 694.08 0.753 176.47 305.91 172.77 0 0 1000 726.47 273.52 3000 1900 450
358.7 358.7 648.6 112.7 138.6 358.8 541.17 800.3 119.86 920.17 727.64 49.306 648.60 840.4 661.7 478.6 49.306 379 104.66 25.62 791.12 694.08 208.84 176.47 305.91 172.77 31.641 17.665 745.86 93.33 273.52 0 0 358.79
ð23Þ
where the bilinear term is the product of the flow rate Fi and the i property operator cp . Then, bounds are established over the variables of the bilinear terms: FLi rF i r FUi
ð24Þ
cip,L r cip r cip,U
ð25Þ
Using these limits, one can write the following linear constraints: i
i
i
i
i
i
i
i
i
i
i
xip Z FLi cp þ cp,L F i FLi cp,L i
xip Z FUi cp þ cp,U F i FUi cp,U xip r FLi cp þ cp,U F i FLi cp,U xip r FUi cp þ cp,L F i FUi cp,L
ð26Þ ð27Þ ð28Þ
ð29Þ
Eqs. (26)–(29) correspond to the convex and concave envelopes of the bilinear terms over the given bounds (Sherali and Alameddine, 1992). In order to tighten the bounds from the LP relaxation, and thus reduce the number of iterations of the branch and bound procedure, one can partition the original domain Di ¼ ½FLi ,FUi of each flow Fi into an arbitrary number of intervals ðDi1 ,Di2 , . . . ,DiN Þ using the points FLi ¼ F1i ,F2i , . . . ,FNi þ 1 ¼ FUi with piecewise linear under estimators for the bilinear terms over each partition. The partition of the domain and the construction
Table 6 Flows for Example 1. Stream
Flow (lb/h)
F1 f1.1 f1.2 w1.COMP w2.COMP w2.TOX w2.NONE dCOMP.TOX dCOMP.NONE dTOX.THOD dTOX.POH dTHOD.POH mCOMP MTOX mTHOD mPOH mNONE gCOMP.1 gCOMP.2 gTOX.3 gPOH.3 gNONE.1 gNONE.2 GS1 GS2 Gwaste
513.5 377.6 135.8 2900 189.1 8.485 2252.3 955.0 126.6 20.85 1.001 20.85 3089.1 963.5 20.85 21.85 2379 1003.6 1003.8 941.6 21.85 1618.7 760.3 3000 1900 963.51
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of the piecewise estimators can be set using the following disjunctions: 2
xip r i
7 6 i i i 7 6 xip ZF i cp,L þ Fni cp Fni cp,L 7 6 7 6 6 xi Z F i ci þF i ci F i ci 7 6 p p,U nþ1 p n þ 1 p,U 7 7 _ 6 6 xi r F i ci þF i ci F i ci 7 6 p p,L nþ1 p n þ 1 p,L 7 7 6 n ¼ 1...N6 7 i i i 6 xip r F i cp,U þ Fni cp Fni cp,U 7 7 6 7 6 Fni rF i r Fni þ 1 7 6 5 4 cip,L r cip r cip,U
ð36Þ
i
Fni ln r pin r Fni þ 1 ln
ð37Þ
cip,L lin r wip,n r cip,U lin
ð38Þ
The partition is done only on the flow rates and not on the properties, in order to avoid a significant increase in the number of variables (both continuous and binary.) This is a key advantage that enables the computational algorithm to be efficient regardless of the number of targeted properties. As noticed by Karuppiah and Grossmann (2006), when the number of partitions increases, there is a trade-off between the computational effort and the relaxation tightness. The resulting model, which involves Eqs. (1)–(38), is a relaxed mixed integer linear programming (MILP) problem. Global optimization algorithm: The steps of the global optimization algorithm are as follows (Quesada and Grossmann, 1995; Karuppiah and Grossmann, 2006):
where lin is a logical variable; when it is true the constraints in the nth disjunction are enforced, and the rest of the constraints are neglected. The convex hull reformulation for this set of disjunctions is as follows (Karuppiah and Grossmann, 2006): X i ln ¼ 1 ð30Þ
Step 1. Preprocessing: Determine the bounds for the variables
n
Fi ¼
pin cip,U þ Fni wip,n Fni cip,U lin
n
3
lin
X
4371
X
pin
ð31Þ
n
cip ¼
X
wip,n
ð32Þ
n
xip Z
X
pin cip,L þFni wip,n Fni cip,L lin
ð33Þ
n
xip Z
X
pin cip,U þ Fni þ 1 wip,n Fni þ 1 cip,U lin
ð34Þ
n
xip r
X
pin cip,L þFni þ 1 wip,n Fni þ 1 cip,L lin
ð35Þ
n
by physical inspection or data given by the problem. Solve the original non-convex MINLP to obtain an initial overall upper bound (OUB) on the objective function. Any feasible solution (e.g., a local minimum solution obtained via a local optimization solver) provides a valid upper bound. Step 2. Lower bounding: Solve the relaxed MILP over a given subregion (the initial subregion is the entire feasible region) to obtain a valid lower bound. If the relaxed problem is infeasible or greater than the current OUB, discard that region. Step 3. Compare the upper and lower bounds. If tolerance is not achieved, select a branching variable. In this case, use only flow rates as branching variables. Although the bound contraction can be performed on the property operators, such strategy it is not implemented because of the large number of operators
F1
Fresh1
INTERCEPTORS MIX PROCESS w1, COMP SOURCES W1 1 w2, COMP
mCOMP
COMP 2
f1, 2 SPLIT
dCOMP, TOX
MIX
mTOX
TOX 1
dTOX, THOD MIX
mTHOD
gCOMP, 1 MIX
gCOMP, 2
Sink 1
gTOX, 3
dTOX, POH
MIX THOD 2
GS1
SPLIT GS2
Sink 2
SPLIT gNONE, 2
w2, TOX W2
f1, 1
gNONE, 1
dTHOD, POH 2 MIX
mPHO
POH 1
SPLIT gPOH, 3
w2, NONE
dCOMP, NONE MIX
mNONE
NONE
SPLIT
Fig. 5. Optimal solution for Example 1.
MIX
Gwaste
Waste
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Table 7 Properties of the streams entering the sinks for the solutions to the examples. Sink
Comp. (ppm)
Toxicity (%)
THOD (mg O2/l)
pH
Density (lb/l)
Viscosity (cP)
Example 1 1 2 Waste
0.013 0.011 0.005
0.539 0.619 0
72.489 75 75
6.144 5.939 5.8
2.132 2.095 2.013
1.202 1.221 1.253
Example 2 1 2 3 Waste
0.013 0.011 0.018 0.005
0.479 0.689 0.34 0
23.286 39.377 9.879 63.529
6.359 6.065 6.309 5.677
2.114 2.135 2.038 2.22
1.202 1.247 1.249 1.247
Example 3 1 2 3 4 5 6 Waste
0.097 0.01 0.04 0.02 0.01 0.01 0.005
0.644 0.837 0.773 0.815 0.837 0.837 0
86.683 94.417 93.013 93.791 94.417 94.417 50
5.8 5.5 5.4 5.5 5.5 5.5 5.4
2.252 2.143 2.18 2.155 2.143 2.143 2.136
1.224 1.241 1.239 1.24 1.241 1.241 1.245
involved with each stream, which would significantly increase the computational time to solve the problem. Step 4. Solve the MINLP, and update OUB if there is an improvement. Step 5. Repeat steps 2–4 for any active subregion until the upper and lower bounds are less or equal to a given tolerance. If the tolerance is reached or there are no subregions left, the global solution corresponds to the best upper bound. It is also important to state that the branching variable (flow rate) is selected according to the following rules: (1) Choose the flow rate that produces the highest difference between the MINLP and MILP problems. (2) If no further improvement is achieved with the previous rule, or if the same variable occurs, select the flow rate that produces the second highest difference, and so on.
250000
200000
150000
100000
OUB LB
50000 1
2
3
4
5
6
7
8
Fig. 6. Evolution of the global optimization algorithm for Example 1.
4. Case studies Three examples were solved to show the application of the proposed methodology. An operation time of 8000 h/yr was considered for all of the examples. The properties assumed to be treated were composition, toxicity, THOD, and pH; the efficiency and cost of each available treatment units are given in Table 2. Example 1. A motivational example is first considered with two process sources, two process sinks and one fresh source. Table 3 presents the characteristics of the process and fresh sources, and Table 4 gives the process and environmental constraints for this problem. The fresh source cost is $0.009/lb. The solution in the root node for the MINLP yields an objective function value of $247,524/yr, which corresponds to the overall upper bound (OUB). The solution for the relaxed model gives an objective function of $97,489/yr. The objective now is to reduce the gap between both solutions by decreasing the objective function value of the MINLP problem and by increasing the objective function value of the relaxed (MILP) problem. To accomplish this task, different branching variables can be selected; as mentioned above, only flow rates are used here as branching variables because of the large number of partitions that property operators would require. From the results of the root node, the first branching variable was taken as the flow rate from the process source w2 to the NONE (or fictitious) unit because this variable gave the highest difference in flow rate values
from all the streams in the network (see Table 5). The domain is partitioned into two subregions, the first one corresponding to the interval 0 rwW2,NONE r 2379:003 lb=h, and the second one to the remaining interval of 2379:003 r wW2,NONE r2450 lb=h. For the first subregion the total annual cost for the MINLP turned out to be $247,524/yr, whereas for the MILP problem the value was $97,321/yr. For the second subregion both problems are infeasible. As can be seen no improvement in the solution could be achieved by using wW2,NONE as branching variable, so the total flow rate entering the toxicity treatment unit ðmTOX Þ was selected as the new branching variable because it yields the second highest difference in flow rate values (see Table 5). For this new branching variable it was found that the solutions for the subregion 840:435r mTOX r 5450 lb=h yielded total annual costs of $276,461/yr and $115,510/yr for the rigorous and relaxed model, respectively, while the analysis for the complementary subregion 0 rmTOX r840:435 lb=h did not provide any improvement of the solutions. From this procedure, the OUB is kept as $247,524/yr because it is the best value up to this point, and the lower bound is updated to the best relaxed solution, which is $115,510/yr. These steps were repeated until the best possible solution for the MINLP was detected, which had an objective function value of $205,908/yr, while that for the MILP was $205,884/yr; the gap between these solutions is 0.012%, and the optimal network is shown in Fig. 5; the flows for the optimal solution are given in
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Table 8 Comparison between simultaneous, sequential, direct recycle and simplified recycle/reuse approaches for Example 1. Concept
Optimal solution (simultaneous approach)
Sequential approach
Direct recycle
Simplified recycle/reuse
F1 (lb/h) Waste (lb/h) Fresh sources cost ($/yr) Treatment units cost ($/yr) Piping cost ($/yr) Network cost ($/yr) Waste treatment system ($/yr) Total annual cost ($/yr)
513.5 963.5 36,973 158,764 10,170 205,908 – 205,908
515.3 965.3 37,108 72,611 9,939 119,660 152,146 271,806
2,268 2,718 163,309 – 7,517 170,826 358,800 529,626
268.16 718.16 19,308 150,180 24,652 194,140 – 194,140
F2
FRESH 2
f2, 1 MIX
GS1
SINK 1
gCOMP, 1 INTERCEPTORS MIX PROCESS SOURCES W1
1
mCOMP
COMP 1
SPLIT
gCOMP, 2
dCOMP, TOX
w2, COMP w1, TOX
MIX
mTOX
TOX 1
GS2 MIX
gNONE, 2
SPLIT
w3, COMP W2
W3
MIX
GS3 SINK 3
gNONE, 1
2
3
SINK 2
w2, NONE
MIX
mNONE
gTOX, 4 NONE
SPLIT
gNONE, 3 MIX
Gwaste
WASTE
Fig. 7. Optimal solution for Example 2.
700000
Table 9 Flows for Example 2. Stream
Flow (lb/h)
F2 f2.1 w1.TOX w1.NONE w2.COMP w2.NONE w3.COMP dCOMP.TOX mCOMP mTOX mNONE gCOMP.1 gCOMP.2 gTOX.4 gNONE.1 gNONE.2 gNONE.3 GS1 GS2 GS3 Gwaste
977.5 977.5 536.04 7546.9 2023.0 1876.9 3279 2813.5 5302.0 3349.5 9,423.8 1370.8 1117.6 3349.5 3651.5 1372.3 4400 6000 2490 4400 3349.5
Table 6. As can be seen in Table 7, the sets of process and environmental constraints are both satisfied; notice that environmental restrictions are near their upper or lower
600000 500000 400000 300000 200000 OUB 100000
LB
0 0
2
4
6
8
10
Fig. 8. Evolution of the global optimization algorithm for Example 2.
bounds. Fig. 6 shows the evolution of the global optimization algorithm. We found that six intervals or partitions of the original flows domain (given by the difference between their original upper and lower bounds) yielded good results. In this example all of the treatment units were used. To treat composition and THOD the cheapest and least efficient units were selected, but for
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toxicity and acidity the most efficient units were needed in order to satisfy the environmental constraints. The same problem was solved in a sequential manner, running first the model without environmental constraints, and then using property treatment systems as needed to meet such constraints. The total annual cost of the process network optimized under the first part of this sequential solution was $119,660/yr; the waste obtained under this scenario did not satisfy the environmental constraints for the following properties: composition (0.013 ppm), toxicity (0.671%), THOD (80.578 mg O2/lt), and pH (5.225). The cost to treat these properties to meet the environmental constraints turned out to be $152,147/yr, which yielded a total annual cost of $271,807/yr. This value is 32% higher than that obtained using the simultaneous approach proposed in this work.
This problem was also solved using the methodologies proposed by Ponce-Ortega et al. (2009, 2010) in order to compare the results with a direct recycle approach and a simplified recycle/ reuse approach. The solutions here presented illustrate the expected trends due to the typical behavior of each model. The direct recycle strategy had to be adjusted to obtain a comparable result, with the main modification consisting in the addition of restrictions over the properties in the process sinks since Ponce-Ortega et al. (2009) only considered constraints over the properties in the waste discharged to the environment. This modification yielded a solution with significantly higher fresh water consumption needed to satisfy process constraints, in addition to the increase in the cost of the treatment units because of the amount of waste generated. The total annual cost for the direct recycle strategy was $529,626/yr, with a fresh water consumption of 2268 lb/h. This solution uses 342% more fresh water than the proposed methodology because the direct recycle strategy does not consider the regeneration of process streams. The simplified recycle and reuse strategy reported by Ponce-Ortega et al. (2010) yields a total annual cost by $194,140/yr. However, it should be noted that this conceptual approach requires six treatment units, as compared to four for the solution obtained here. The reason is that in Ponce-Ortega et al. (2010) the structure of the property interceptor network is such that streams from process sources are segregated and each branch is treated for one property without recycle, such that each output goes to the process sinks or to the waste stream. As a consequence a high number of units will be promoted as part of the network structure in exchange for a more manageable model that avoids most of the bilinear terms of the one presented in this work. Table 8 lists the relevant results from these methodologies.
Table 10 Comparison between simultaneous and sequential approaches for Example 2. Concept
Optimal solution (simultaneous approach)
Sequential approach
F2 (lb/h) Waste (lb/h) Fresh sources cost ($/yr) Treatment units cost ($/yr) Piping cost ($/yr) Network cost ($/yr) Waste treatment system ($/yr) Total annual cost ($/yr)
977.5 3349.5 46,924 538,316 29,297 614,538 – 614,538
879.8 3251.8 42,232 155,504 28482 226,219 424,039 65,258
F3 PROCESS SOURCES W1 1
MIX
w1.COMP W2
2 w2.TOX
w3.COMP
INTERCEPTORS MIX
w2.NONE W3
FRESH 3
mCOMP
MIX COMP 1
w3.NONE
MIX
w4.COMP
mTOX
TOX 1
SPLIT
THOD 2
SPLIT
GS3
SINK 3
dTOX.THOD
4 MIX w6.COMP
mTOX
MIX
GS4
SINK 4
dTHOD.POH MIX
5 w5.NONE
W6
SINK 2
dCOMP.TOX MIX
W5
GS2
SINK 1
SPLIT
3
W4
GS1
MIX
mTOX
mNONE
POH2
SPLIT
NONE
SPLIT
6
MIX
MIX
MIX Fig. 9. Optimal solution for Example 3.
GS5
GS6
GS7
SINK 5
SINK 6
WASTE
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Example 2. In this example we consider the integration problem with three process sources and three process units; two fresh sources are available with unit costs of $0.009/lb and $0.006/lb. Table 3 presents the characteristics for the process and fresh sources, whereas Table 4 gives the process and environmental constraints that must be satisfied. After exhausting every open node, the optimal solution was found with a minimum gap between MINLP and relaxed solutions of 4.4%. The total annual cost of the best network was $614,539/ yr, with the configuration shown in Fig. 7 and the flow structure given in Table 9. Table 7 reports the sink and waste properties obtained for this solution. In this case, only two treatment units, one for component recovery and the other one for toxicity, were needed to meet the environmental constraints; both units were the ones with maximum efficiency. Process sources 2 and 3 are mixed to be treated for composition. Process source 1 is split, a fraction of this source is treated for toxicity and the rest is mixed with a fraction of process source 2. The mixture of process sources 1 and 2 is sent directly to the process sinks without being treated by the property interceptors. As far as the fresh sources, only source 2 (the cheapest choice) was used. The evolution of the objective function values for the MINLP and the relaxed models for each of the 10 iterations needed to solve this problem can be observed in Fig. 8. In this example, 12 partitions of the domain were used. Table 10 compares the result from the solution obtained with the proposed method to the solution obtained with a sequential approach similar to the one used in Example 1. It can be seen how in this case the sequential solution yields a final network with a total annual cost 5.8% higher than the simultaneous solution. Example 3. This example contains six process sources, six sinks and three fresh sources. Process and fresh sources data are given in Table 3. Table 4 presents the sinks and waste constraints. The cost of the fresh sources 1, 2 and 3 are $0.008/lb, $0.006/lb and $0.003/lb, respectively. The initial solution for the root node was obtained, and after visiting each open node it was found that no improvement to the initial solution could be detected. The optimal solution therefore corresponds to one of the root node, with a total annual cost of $487,746/yr and a minimum gap between MINLP and relaxed solutions of 6.52%. The optimal network is shown in Fig. 9, and the flow rate distribution for the integrated process is given in Table 11. All the constraints are satisfied (see Table 7), and as in the previous examples the properties were near their maximum or minimum values. The network structure shows a noticeable use of the property interceptors. Process sources 1, 3, 4 and 6 are treated for composition; 477 lb/h of the output from the composition interceptor (equivalent to 6% of the total output) is mixed with 10 lb/h of process source 2 and treated for toxicity; from the output of the toxicity interceptor, 425 lb/h are sent for THOD treatment, and 61 lb/h for POH treatment. The total amount of process source 5 (1300 lb/h) together with some splits of process sources 2 and 3 bypass the use of the property interceptors and go directly to the process sinks. A sequential solution was also implemented for this case, and, as indicated in Table 12, the solution using the sequential approach gave an initial network with an annual cost 29% lower than the simultaneous approach, but when the final treatment system was added to the sequential solution for a complete network that meets the environmental constraints, the network cost increased to $751,147/yr, or 54% higher than the simultaneous approach. For the sequential approach it can also be seen that the fresh water consumption was higher than in the network obtained with the simultaneous model. For the solution with
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Table 11 Flows for Example 3. Stream
Flow (lb/h)
F3 f3.1 f3.2 f3.3 f3.4 f3.5 f3.6 w1.COMP w2.TOX w2.NONE w3.COMP w3.NONE w4.COMP w5.NONE w6.COMP dCOMP.TOX dTOX.THOD dTOX.POH mCOMP mTOX mTHOD mPOH mNONE gCOMP.2 gCOMP.3 gCOMP.4 gCOMP.5 gCOMP.6 gTOX.7 gTHOD.7 gPOH.7 gNONE.1 gNONE.2 gNONE.3 gNONE.4 gNONE.5 gNONE.6 GS1 GS2 GS3 GS4 GS5 GS6 Gwaste
237.79 93.52 25.95 38.62 34.27 29.19 16.22 3100 10.234 1789.7 1378.1 371.88 2000 1300 1400 477.55 425.19 61.09 7878.1 487.79 425.19 61.09 3461.64 1478.67 1697.9 1636.2 1663.5 924.17 62.59 364.09 61.098 1806.4 95.36 1063.4 329.48 107.28 59.60 1900 1600 2800 2000 1800 1000 487.7
Table 12 Comparison between simultaneous and sequential approach for Example 3. Concept
Optimal solution (simultaneous approach)
Sequential approach
F1 (lb/h) F3 (lb/h) Waste (lb/h) Fresh sources cost ($/yr) Treatment units cost ($/yr) Piping cost ($/yr) Network cost ($/yr) Waste treatment system ($/yr) Total annual cost ($/yr)
– 237.79 487.7 5707 460,451 21,587 487,746 – 487,746
1800.2 166.09 2216.3 119,200 205,123 20,792 345,116 406,030 751,146
Table 13 Computational effort for the case studies. Example
Number of process sources
Number of process sinks
Number of partitions
Time (s)
1 2 3
2 3 6
2 3 6
6 12 33
131.29 908.62 4505.17
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Table 14 Computer benchmark using Linpacks java version. Benchmark Millions of floating point operations per second Time Norm res Precision
331.57 Mflops/s 0.25 s 5.68 2.2204 10 16
fr,j Fr gu,j Gj HY mu mIðUÞ Uðp0 Þ MCost
the simultaneous approach, the selected units for component and toxicity treatment were the ones with maximum efficiency, but treatment for THOD and pH was satisfactorily met using the units with the lowest efficiency. For this example 32 intervals were used as partitions of the original flows domain. Computational effort: The time needed to solve each of the examples in a computer with an Intels CoreTM2 CPU T5200 at 1.6 GHz and 2.00 GB RAM is reported in Table 13. This computer was benchmarked using the java version of the linpack software (available at http://www.netlib.org/benchmark/linpackjava/), and the results are summarized in Table 14. It can be seen how the computational effort depends directly on the size of the problem, and more noticeably on the number of partitions of the original flows domain.
Mm M cp NFresh Nint Nproperties NSinks NSources NUnits pH pipExit u,j pipin i,u
5. Conclusions pipinFresh r,j This paper has presented a mathematical representation of a mass and property integration using a recycle–reuse scheme for a network that includes process interceptors inside of the process. The formulation has considered simultaneously both process and environmental constrains for properties such as composition, toxicity, THOD, pH, density and viscosity. A set of disjunctions were formulated for the use of in-plant property interceptors. The resulting MINLP model for the mass integration network contains several bilinear terms, and it was solved using a global optimization strategy, which involves the solution of an MINLP model and a relaxed MILP model to provide under- and over-estimations of the objective function. A branch and bound strategy was used until the estimators approached each other within a minimum gap. Several partitions of the flows domain were implemented as part of the solution procedure. The solution to three examples has shown the application of the proposed methodology, and its advantage over a typical sequential approach of integrating a process network first and then adding the property treatment systems to meet the environmental constraints. A comparison has also been presented with previous works in which a centralized treatment system (direct recycle strategy), and a simplified recycle and reuse formulation have been used. It is also shown how the computational effort depends directly on the size of the problem and on the number of partitions chosen to strengthen the bounds of the relaxation procedure. Further work should aim towards the improvement of the property operators to make better predictions of the properties in the mixing points of the network. Also a more detailed design model for the property interceptors should be developed instead of using a constant value for the unit efficiency.
Notation CostrFresh CostuUnit unit,IðUÞ CostUðp 0Þ
unit cost for fresh utility r
du1 ,u
flow rate from treatment unit u1 to treatment unit u
unit cost for treatment unit u disaggregated variable for CostuUnit
pipint u1 ,u TAC THOD Tox wi,u waste Wi xip 0
Uðp Þ yIðUÞ
z
segregated flow rate from fresh source r to sink j total flow rate for fresh source r segregated flow rate from treatment unit u to process sink j total flow rate for process sink j plant operating time, hours per year total flow rate entering unit u disaggregated variable for mu upper bound used in the convex hull formulation for the cost of the treatment unit upper bound used in the convex hull formulation for the flow rate entering the treatment units upper bound used in the convex hull formulation for the property operator total number of fresh sources total number of units available to treat each property total number or properties total number of sinks total number of process sources total number of treatment units potential of hydrogen piping cost to send segregated flow rate from treatment unit u to process sink j piping cost to send process source i to treatment unit u piping cost to send fresh source r to treatment process sink j piping cost to send segregated flow rate from treatment unit u1 to treatment unit u total annual cost theoretical oxygen demand toxicity segregated flow rate from process source i to treatment unit u total flow rate for the waste stream discharged to the environment total flow rate for process source i variable that substitutes the product flow rate i times the property operator of the property p binary variable used to select the appropriate treatment unit for each property composition
Sets FRESH NPROP NINT
set for fresh sources, frjr ¼ 1, . . . ,NFresh g set for the properties, fpjp ¼ 1,:::,Nproperties g set for the available units for each property, fIjI ¼ 1, . . . ,Nint g NSINKS set for sinks, fjjj ¼ 1, . . . ,NSinks g NSOURCES set for process sources, fiji ¼ 1, . . . ,NSources g NUNITS set for the treatment units, fuju ¼ 1, . . . ,NUnits g
Greek letters 0
Uðp Þ aIðUÞ
efficiency of property interceptor for property p
lin
binary variable used to select the active region in the domain partition viscosity disaggregated variable for the flow rate i used in the domain partition density disaggregated variable for the property operator p used in the domain partition property operator for the mixing rule for property p
m pin r wip,n cp
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Indices i In j max min n Out p r Sink Source u waste
process source inlet sink maximum minimum interval out property fresh source sink source unit waste discharged to the environment
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